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The 1st a part of this examine monograph discusses basic houses of G-ENRBs - Euclidean Neighbourhood Retracts over B with motion of a compact Lie crew G - and their relatives with fibrations, non-stop submersions, and fibre bundles. It therefore addresses equivariant aspect set topology in addition to equivariant homotopy conception.

5. 6. 31 are σ-conjugate to the identity element. Therefore in the classical case there is no need to introduce the parameter b which we see above. On the other hand, the Lang map G(L) → G(L) is not surjective. In fact, we have seen above that G(L) consists of many σ-conjugacy classes. The usual proof for the surjectivity of the Lang map fails in the setting of ind-schemes: although the diﬀerential of the Lang map g → g −1 σ(g) is an isomorphism, one cannot conclude that the map itself is “´etale”.

6. It is expected that in the hyperspecial case, all aﬃne Deligne–Lusztig varieties are equidimensional. 1 in [26] shows that Xμ (b) is equidimensional. The proof relies on a result proved by Mirkovi´c and Vilonen as part of their proof of the geometric Satake isomorphism. More precisely, their results about the intersection cohomology of intersections of U (L)and K-orbits imply that these intersections are equidimensional. For a proof of the relevant fact in positive characteristic, which is what one needs in our situation, see the paper [62] by Ngˆ o and Polo.

2. The σ-conjugacy class can be represented by an element τ ∈ W with (τ ) = 0. We call an element b ∈ G(L) basic, if its σ-conjugacy class is basic. 1). Looking at σ-conjugacy classes from the point of view of Newton strata in the special ﬁber of a Shimura variety, the basic locus is the unique closed Newton stratum. In the case of the Siegel modular variety, for instance, this is just the supersingular locus. 3. Aﬃne Deligne–Lusztig varieties: the hyperspecial case Similarly as for usual Deligne–Lusztig varieties, we want to consider all elements g which map under a “Lang map” to a ﬁxed double coset.